(ζ−m, ζm)-Type Algebraic Minimal Surfaces in Three-Dimensional Euclidean Space

We introduce the real minimal surfaces family by using the Weierstrass data (ζ−m,ζm) for ζ∈C, m∈Z≥2, then compute the irreducible algebraic surfaces of the surfaces family in three-dimensional Euclidean space E3. In addition, we propose that family has a degree number (resp., class number) 2m(m+1) in the cartesian coordinates x,y,z (resp., in the inhomogeneous tangential coordinates a,b,c).

GENUS MINIMAL k-NOIDS AND SADDLE TOWERS IN

For each $k\geq 3$ , we construct a $1$ -parameter family of complete properly Alexandrov-embedded minimal surfaces in the Riemannian product space $\mathbb {H}^2\times \mathbb {R}$ with genus $1$ and k embedded ends asymptotic to vertical planes. We also obtain complete minimal surfaces with genus $1$ and $2k$ ends in the quotient of $\mathbb {H}^2\times \mathbb {R}$ by an arbitrary vertical translation. They all have dihedral symmetry with respect to k vertical planes, as well as finite total curvature $-4k\pi $ . Finally, we provide examples of complete properly Alexandrov-embedded minimal surfaces with finite total curvature with genus $1$ in quotients of $\mathbb {H}^2\times \mathbb {R}$ by the action of a hyperbolic or parabolic translation.

The intrinsic group–subgroup structures of the Diamond and Gyroid minimal surfaces in their conventional unit cells

The intrinsic, hyperbolic crystallography of the Diamond and Gyroid minimal surfaces in their conventional unit cells is introduced and analysed. Tables are constructed of symmetry subgroups commensurate with the translational symmetries of the surfaces as well as group–subgroup lattice graphs.

Geometric convergence results for closed minimal surfaces via bubbling analysis

We present some geometric applications, of global character, of the bubbling analysis developed by Buzano and Sharp for closed minimal surfaces, obtaining smooth multiplicity one convergence results under upper bounds on the Morse index and suitable lower bounds on either the genus or the area. For instance, we show that given any Riemannian metric of positive scalar curvature on the three-dimensional sphere the class of embedded minimal surfaces of index one and genus $$\gamma $$ γ is sequentially compact for any $$\gamma \ge 1$$ γ ≥ 1 . Furthemore, we give a quantitative description of how the genus drops as a sequence of minimal surfaces converges smoothly, with mutiplicity $$m\ge 1$$ m ≥ 1 , away from finitely many points where curvature concentration may happen. This result exploits a sharp estimate on the multiplicity of convergence in terms of the number of ends of the bubbles that appear in the process.

Partial islands and subregion complexity in geometric secret-sharing model

We compute the holographic subregion complexity of a radiation subsystem in a geometric secret-sharing model of Hawking radiation in the “complexity = volume” proposal. The model is constructed using multiboundary wormhole geometries in AdS3. The entanglement curve for secret-sharing captures a crossover between two minimal curves in the geometry apart from the usual eternal Page curve present for the complete radiation entanglement. We compute the complexity dual to the secret-sharing minimal surfaces and study their “time” evolution. When we have access to a small part of the radiation, the complexity shows a jump at the secret-sharing time larger than the Page time. Moreover, the minimal surfaces do not have access to the entire island region for this particular case. They can only access it partially. We describe this inaccessibility in the context of “classical” Markov recovery.